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arXiv:quant-ph/0108103 · Physical Review A

Nonadiabatic controlled-NOT gate for the Kane solid-state quantum computer

Cameron J. Wellard, Lloyd C. L. Hollenberg, Hans–Christian Pauli

Published Feb 4, 2002• 7 citations

Abstract

The method of iterated resolvents is used to obtain an effective Hamiltonian for neighboring qubits in the Kane solid-state quantum computer. In contrast to the adiabatic gate processes inherent in the Kane proposal we show that free evolution of the qubit-qubit system, as generated by this effective Hamiltonian, combined with single-qubit operations, is sufficient to produce a controlled-NOT gate. Thus the usual set of universal gates can be obtained on the Kane quantum computer without the need for adiabatic switching of the controllable parameters as prescribed by Kane [Nature (London) 393, 133 (1998)]. Both the fidelity and the gate time of this nonadiabatic controlled-NOT gate are determined by numerical simulation.

DOI: 10.1103/physreva.65.032303

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The method of iterated resolvents is used to obtain an effective Hamiltonian for neighboring qubits in the Kane solid-state quantum computer. In contrast to the adiabatic gate processes inherent in the Kane proposal we show that free evolution of the qubit-qubit system, as generated by this effective Hamiltonian, combined with single-qubit operations, is sufficient to produce a controlled-NOT gate. Thus the usual set of universal gates can be obtained on the Kane quantum computer without the need for adiabatic switching of the controllable parameters as prescribed by Kane [Nature (London) 393, 133 (1998)]. Both the fidelity and the gate time of this nonadiabatic controlled-NOT gate are determined by numerical simulation.

In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.

In this paper we investigate the effect of dephasing on proposed quantum gates for the solid-state Kane quantum computing architecture. Using a simple model of the decoherence, we find that the typical error in a controlled-NOT gate is $8.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. We also compute the fidelities of

Z

,

X

, swap, and controlled

Z

operations under a variety of dephasing rates. We show that these numerical results are comparable with the error threshold required for fault tolerant quantum computation.

Unitary quantum gates constitute the building blocks of quantum computing in the circuit paradigm. In this work, we engineer a locally driven two-qubit Hamiltonian whose instantaneous ground-state dynamics generates the controlled-NOT (CNOT) quantum gate. In practice, quantum gates have to be implemented in finite time, hence nonadiabatic and external noise effects debilitate gate fidelities. Here, we show that counterdiabatic control can restore gate performance with near perfect fidelities even in open quantum systems subject to decoherence.

Abstract Unitary quantum gates constitute the building blocks of quantum computing in the circuit paradigm. In this work, we engineer a locally driven two-qubit Hamiltonian whose instantaneous ground-state dynamics generates the controlled-NOT (CNOT) quantum gate. In practice, quantum gates have to be implemented in finite time, hence nonadiabatic and external noise effects debilitate gate fidelities. Here, we show that counterdiabatic control can restore gate performance with near perfect fidelities even in open quantum systems subject to decoherence.

We propose and find pulse sequences for non‐adiabatic two‐qubit gates (CNOT, swap and controlled Z gates) for the Kane silicon‐based electron mediated nuclear spin quantum computer architecture. These quantum gates are simpler, with higher fidelity and faster than existing proposals. Any two qubit gate may be implemented using similar pulse sequences. In this paper, we investigate the effect of decoherence on the operation of these quantum gates and some simple algorithms for a variety of dephasing rates. Our model shows that conservative estimates of the dephasing rates in the Kane architecture give gate fidelities comparable to that required for fault tolerant quantum computing.